Physical Church-Turing Thesis

Geometry in the works of mathematicians during the Islamic era included three basic parts: theoretical geometry, practical geometry, and geometry in astronomy. Theoretical geometry related to the tradition of Greek mathematicians like Euclid. Calculations of the area of lands, dividing land among inheritors, opening roads between lands, mensuration, and so on are examples of practical geometry. There is evidence that artists, carpenters, surveyors, blacksmiths, etc., used practical geometry to create their works during the Islamic era. There are references to the subject of practical geometry in science classification books by Ibn al-Akfānī and Ibn al-Nadīm.

Beginning with Turing's seminal work in 1950, artificial intelligence proposes that consciousness can be simulated by a Turing machine. This implies a potential theory of everything where the universe is a simulation on a computer, which begs the question of whether we can prove we exist in a simulation. In this work, we construct a relative model of computation where a computable local machine is simulated by a global, classical Turing machine. We show that the problem of the local machine computing simulation properties of its global simulator is undecidable in the same sense as the Halting problem. Then, we show that computing the time, space, or error accumulated by the global simulator are simulation properties and therefore are undecidable. These simulation properties give rise to special relativistic effects in the relative model which we use to construct a relative Church-Turing-Deutsch thesis where a global, classical Turing machine computes 1

We think that computational models at the quantum level can solve some problems (at least in the sense of computational complexity) that are hard to calculate by classical models.

We will simply browse that topic considered by many authors, that is, the computational results of computational models with special structures that can send resources back to the past provided by quantum theory and general relativity [24]. These results mainly revolve around Deutschian CTCs and post-selected CTCs based on quantum theory. Compared with this, there are far fewer results for computational results in general relativity, for example, in [7].

This paper considers whether computational formalisms beyond the Church Turing Thesis (CTT) could be helpful in understanding the mind. We argue that they may be, and that the way that the CTT has been invoked in Cognitive Science may therefore act as a Red Herring. That is, the way the CTT is invoked in Cognitive Science may mislead and perhaps contribute to premature abandonment of possibly fruitful research directions in Cognitive Science. We do not suggest some sort of "hypercomputation'. Whilst it is possible to use a rich interactive machine to implement a simple function this does not lead to new computable functions. In other words, the CTT is valid even if more sophisticated machinery is employed. It is the other direction that is the core of this paper: When considering more sophisticated computational tasks, then standard Turing machines (and their mode of operation) are not sufficient to explore the range of possibilities. The CTT is commonly interpreted as stating that the intuitive concept of computability is fully captured by Turing machines or any equivalent formalism (such as recursive functions, the lamba calculus, Post production rules, and many others). The CTT implies that if a function is (intuitively) computable, then it can be computed by a Turing machine. Conversely, if a Turing machine cannot compute a function, it is not computable by any mechanism whatsoever. We suggest an inadvertent error that has been made which is the claim that relatively simple computational formalisms like Turing Machines can do anything that more complex computional formalisms can do. To show this we present the landscape of computability within and beyond the bounds covered by the mathematical CTT. This shows that in regions of the computational landscape beyond the CTT there may be hierarchies of increasingly powerful computational formalisms. Erroneously interpreting CTT as enforcing a 'one size fits all' interpretation to computational formalisms leads to extreme reductionism that means contemporary computationalism is viewed as inadequate to explaining many phenomena related to thought and mind in living systems. Once this Red Herring interpretation for CTT is avoided this leaves the way open to exploring how richer kinds of computation that may possess many shades of expressivity can form part of Cognitive Science explanations.

This paper considers whether computational formalisms beyond the Church Turing Thesis (CTT) could be helpful in understanding the mind. We argue that they may be, and that the way that the CTT has been invoked in Cognitive Science may therefore act as a Red Herring. That is, the way the CTT is invoked in Cognitive Science may mislead and perhaps contribute to premature abandonment of possibly fruitful research directions in Cognitive Science. We do not suggest some sort of "hypercomputation'. Whilst it is possible to use a rich interactive machine to implement a simple function this does not lead to new computable functions. In other words, the CTT is valid even if more sophisticated machinery is employed. It is the other direction that is the core of this paper: When considering more sophisticated computational tasks, then standard Turing machines (and their mode of operation) are not sufficient to explore the range of possibilities. The CTT is commonly interpreted as stating that the intuitive concept of computability is fully captured by Turing machines or any equivalent formalism (such as recursive functions, the lamba calculus, Post production rules, and many others). The CTT implies that if a function is (intuitively) computable, then it can be computed by a Turing machine. Conversely, if a Turing machine cannot compute a function, it is not computable by any mechanism whatsoever. We suggest an inadvertent error that has been made which is the claim that relatively simple computational formalisms like Turing Machines can do anything that more complex computional formalisms can do. To show this we present the landscape of computability within and beyond the bounds covered by the mathematical CTT. This shows that in regions of the computational landscape beyond the CTT there may be hierarchies of increasingly powerful computational formalisms. Erroneously interpreting CTT as enforcing a 'one size fits all' interpretation to computational formalisms leads to extreme reductionism that means contemporary computationalism is viewed as inadequate to explaining many phenomena related to thought and mind in living systems. Once this Red Herring interpretation for CTT is avoided this leaves the way open to exploring how richer kinds of computation that may possess many shades of expressivity can form part of Cognitive Science explanations.

decided to participate in Savannah Law Review's symposium, Rise of the Automatons, because of the call for papers, which began with a quote attributed to John von Neumann: "You insist that there is something a machine cannot do. If you tell me precisely what it is a machine cannot do, then I can always make a machine which will do just that." Two paragraphs later, the CFP refers to the coming technological Singularity, "the idea that A.I. will improve itself into a Superintelligence that surpasses the capacity of collective human intelligence." John von Neumann was simply one of the smartest people living in the twentieth century. He died young and never completed his own project of theorizing and building a self-reproducing machine. (See infra note 46.) I am, well, just me. If I want to say there are aspects of automaton-lawyering we should consider in constructing legal education curricula, I feel obliged to dig into the technical weeds. If the essay contributes a "go to" source in the law reviews for a technical explanation of the Halting Problem, all the better.

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Most Computer Science faculty members loo k forward to teaching a course in Theory o f Computation. However, there is usuall y concern over how well students are able t o understand Turing's thesis, Church' s thesis, and the relationship between the m with only one semester of concentration o n details of a particular textbook. Thi s paper describes how students in the Theor y of Computation Course at Denison Universit y designed their own theoretic machines and showed how those machines could be used t o solve example problems, hopefully achievin g an understanding of the theses by Churc h and Turing as a result of developing th e technical details necessary in describin g their machines. Course Conten t The students used the textbook Theory o f Computation : Formal Languages, Automata , and Complexity by J. Glenn Brookshear. This text covers the usual topic s finite state machines, pushdown automata , Turing machines, grammars, recursiv e functions, computability and complexity .

Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using as a springboard Gödel's little-known assertion that the human mind has a power "converging to infinity," and as an anchoring problem Rado's (1963) Turing-uncomputable "busy beaver" (or Σ) function, we present in this short paper a new argument that, in fact, human persons can hypercompute. The argument is intended to be formidable, not conclusive: it brings Gödel's intuition to a greater level of precision, and places it within a sensible case against computationalism.

Computationalism is a relatively vague term used to describe attempts to apply Turing's model of computation to phenomena outside its original purview: in modelling the human mind, in physics, mathematics, etc. Early versions of computationalism faced strong objections from many (and varied) quarters, from philosophers to practitioners of the aforementioned disciplines. Here we will not address the fundamental question of whether computational models are appropriate for describing some or all of the wide range of processes that they have been applied to, but will focus instead on whether 'renovated' versions of the new computationalism shed any new light on or resolve previous tensions between proponents and skeptics. We find this, however, not to be the case, because the new computationalism falls short by using limited versions of "traditional computation", or proposing computational models that easily fall within the scope of Turing's original model, or else proffering versions of hypercomputation with its many pitfalls.

In this party, we want to discuss some of the points when two computational theoretic framework models working on the set of real numbers are the Blum-Shub-Smale machine and the type 2 Turing machine. The focus will be on three points. The first is related to the mathematical model frameworks themselves. In the second, we will briefly show complexity results with an algebraic model of computation (that is, the Blum-Shub-Smale machine). The variants of the Blum-Shub-Smale machine and the type 2 Turing machine obtained will lead us to the final point.

We hope these machines serve well for knowing how to further enrich the universe of oracle machines (or say, demons). On the other hand, for some approaches, we decide to ignore them in the party, for example, the models of analog computations, like the model proposed by Shannon [Sha41] and the series of follow-up works. As same as the computability of the wave equation [PER81], [WZ02] and the computability of Banach space [PER89].

The characterizations of the Church-Turing thesis in real models have their origins in [Pég16].

"Church's thesis" is at the foundation of computer science. We point out that with any particular set of physical laws, Church's thesis need not merely be postulated, in fact it may be decidable. Trying to do so is valuable. In Newton's laws of physics with point masses, we outline a proof that Church's thesis is false; phsyics is unsimulable. But with certain more realistic laws of motion, incorporating some relativistic effects, the Extended Church's thesis is true. Along the way we prove a useful theorem: a wide class of ordinary differential equations may be integrated with "polynomial slowdown." Warning: we cannot give careful definitions and caveats in this abstract-you must read the full text-and interpreting our results is not trivial.

This paper concerns Alan Turing's ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the "mathematical objection" to his view that machines can think. Logico-mathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do.

The Physical Computability Thesis (PCT) states that the physical world is computable. Sometimes it is argued that a well-evidenced logical principle, the Church-Turing Thesis, entails PCT. But this reasoning is faulty. I argue that it is an open question whether PCT is true: even if the universe is finite, physics may turn out to confound PCT. What would a non-computable physics look like, and what would be the implications for scientists and engineers? I review potential countermodels to various formulations of PCT.

Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using as a springboard Gödel's little-known assertion that the human mind has a power "converging to infinity," and as an anchoring problem Rado's (1963) Turing-uncomputable "busy beaver" (or Σ) function, we present in this short paper a new argument that, in fact, human persons can hypercompute. The argument is intended to be formidable, not conclusive: it brings Gödel's intuition to a greater level of precision, and places it within a sensible case against computationalism.

The problem of attempting to learn the mapping between data and labels is the crux of any machine learning task. It is, therefore, of interest to the machine learning community on practical as well as theoretical counts to consider the existence of a test or criterion for deciding the feasibility of attempting to learn. We investigate the existence of such a criterion in the setting of PAC-learning, basing the feasibility solely on whether the mapping to be learnt lends itself to approximation by a given class of hypothesis functions. We show that no such criterion exists, exposing a fundamental limitation in the decidability of learning. In other words, we prove that testing for PAC-learnability is undecidable in the Turing sense. We also briefly discuss some of the probable implications of this result to the current practice of machine learning.

We investigate several unconventional models of finite automata and algorithms. We show that two-way alternating automata can be smaller than fast bounded-error probabilistic automata. We introduce ultrametric finite automata which use p-adic numbers to describe the branching process of the computation. We examine the size complexity of all the abovementioned automata for the counting problem. We also examine two-way frequency finite automata. We define ultrametric query algorithms and examine ultrametric query complexity for Boolean functions. We generalize the notion of frequency computation by requiring some structure for the correct outputs.

This paper considers whether computational formalisms beyond the Church Turing Thesis (CTT) could be helpful in understanding the mind. We argue that they may be, and that the way that the CTT has been invoked in Cognitive Science may therefore act as a Red Herring. That is, the way the CTT is invoked in Cognitive Science may mislead and perhaps contribute to premature abandonment of possibly fruitful research directions in Cognitive Science. We do not suggest some sort of "hypercomputation'. Whilst it is possible to use a rich interactive machine to implement a simple function this does not lead to new computable functions. In other words, the CTT is valid even if more sophisticated machinery is employed. It is the other direction that is the core of this paper: When considering more sophisticated computational tasks, then standard Turing machines (and their mode of operation) are not sufficient to explore the range of possibilities. The CTT is commonly interpreted as stating that the intuitive concept of computability is fully captured by Turing machines or any equivalent formalism (such as recursive functions, the lamba calculus, Post production rules, and many others). The CTT implies that if a function is (intuitively) computable, then it can be computed by a Turing machine. Conversely, if a Turing machine cannot compute a function, it is not computable by any mechanism whatsoever. We suggest an inadvertent error that has been made which is the claim that relatively simple computational formalisms like Turing Machines can do anything that more complex computional formalisms can do. To show this we present the landscape of computability within and beyond the bounds covered by the mathematical CTT. This shows that in regions of the computational landscape beyond the CTT there may be hierarchies of increasingly powerful computational formalisms. Erroneously interpreting CTT as enforcing a 'one size fits all' interpretation to computational formalisms leads to extreme reductionism that means contemporary computationalism is viewed as inadequate to explaining many phenomena related to thought and mind in living systems. Once this Red Herring interpretation for CTT is avoided this leaves the way open to exploring how richer kinds of computation that may possess many shades of expressivity can form part of Cognitive Science explanations.

International audienceComputational complexity theory (CCT) is usually construed as the mathematical study of the complexity of computational problems. In recent years, research work on unconventional computational models has called into question the purely mathematical nature of CCT, and has revealed potential relations between its subject matter and empirical sciences. In particular, recent debates surrounding quantum computing have raised the possibility of a new computational model, one based on quantum mechanics, which may be exponentially more efficient than any previously known machine. In this paper, I will show how those recent debates suggest an alternative , empirical view of CCT. I will then examine what are the fundamental arguments in the favor of this view, what are its consequences for our conception of CCT as a scientific field, and how further philosophical investigation should proceed

I explore the conceptual foundations of Alan Turing's analysis of computability, which still dominates thinking about computability today. I argue that Turing's account represents a last vestige of a famous but unsuccessful program in pure mathematics, viz., Hilbert's formalist program. It is my contention that the plausibility of Turing's account as an analysis of the computational capacities of physical machines rests upon a number of highly problematic assumptions whose plausibility in turn is grounded in the formalist stance towards mathematics. More speciÿcally, the Turing account con ates concepts that are crucial for understanding the computational capacities of physical machines. These concepts include the idea of an "operation" or "action" that is "formal," "mechanical," "well-deÿned," and "precisely described," and the idea of a "symbol" that is "formal," "uninterpreted," and "shaped". When these concepts are disentangled, the intuitive appeal of Turing's account is signiÿcantly undermined. This opens the way for exploring models of hypercomputability that are fundamentally di erent from those currently entertained in the literature.

In their recent paper ''Do Accelerating Turing Machines Compute the Uncomputable?'' Copeland and Shagrir (Minds Mach 21:221-239, 2011) draw a distinction between a purist conception of Turing machines, according to which these machines are purely abstract, and Turing machine realism according to which Turing machines are spatio-temporal and causal ''notional'' machines. In the present response to that paper we concede the realistic aspects of Turing's own presentation of his machines, pointed out by Copeland and Shagrir, but argue that Turing's treatment of symbols in the course of that presentation opens the door for later purist conceptions. Also, we argue that a purist conception of Turing machines (as well as other computational models) plays an important role not only in the analysis of the computational properties of Turing machines, but also in the philosophical debates over the nature of their realization.

The aim of this paper is to present the origin of Church's thesis and the main arguments in favour of it as well as arguments against it. Further the general problem of the epistemological status of the thesis will be considered, in particular the problem whether it can be treated as a definition and whether it is provable or has a definite truth-value.

Peter Wegner's definition of computability differs markedly from the classical term as established by Church, Kleene, Markov, Post, Turing et al.. Wegner identifies interaction as the main feature of today's systems which is lacking in the classical treatment of computability. We compare the different approaches and argue whether or not Wegner's criticism is appropriate. Taking into account the major arguments from the literature, we will show that Church's thesis still holds.

mathematical entities don’t seem to be the right kinds of things to implement computation. Time and change are essential to implementing a computation: computation is a process that unfolds through time, during which the hardware undergoes a series of changes (flip-flops flip, for example, neurons fire and go quiet, plastic counters appear and disappear on a Go board, and so on). Yet Tegmark’s mathematical objects exist timelessly and unchangingly. What plays the role of time and change for this 'hardware'? How could these Platonic objects change over time in order to implement distinct computational steps? And how could one step give rise to the next if there is no time or change? Unchanging mathematical objects are just not the right kinds of things to implement a computation. Currently, there are no plausible solutions to this chicken-and-egg implementation problem. Perhaps supporters of Zuse’s thesis could say: we know that something must implement the universe’s computa...

In this review article we discuss three Hypercomputing models: Accelerated Turing Machine, Relativistic Computer and Quantum Computer based on three new discoveries: Superluminal particles, slowly rotating black holes and adiabatic quantum computation, respectively.

Hilbert, Turing and the idea of effective procedure. Hilbert put forward the properties that an effective procedure should possess, without a formal characterization. Nevertheless, he formulated four desiderata: (1) a calculus is a set of instructions to be executed to solve a problem; 2) it must boil down to rules for the manipulation of formulae of a suitable formal language; 3) it must guarantee the solution of the problem in a finite number of steps; 4) it should be possible to set in advance an upper bound to the number of steps it will take to find the solution. Turing proposed his famous machines to define rigorously the meaning of effective procedure and, paradoxically, showed there is no mathematical characterization of it that satisfies Hilbert’s expectations. In particular, Turing’s model goes beyond desiderata third and fourth. Key words: Effective Procedure, Entscheidungsproblem, Halting Problem, Turing machines.

A version of the Church-Turing Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, non-well-founded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard Church-Turing Thesis.

We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit 8 expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory 9 within the setting of formal mathematical theories.

Nowadays the work of Alan Turing and Kurt Gödel rightly takes center stage in discussions of the relationships between computability and the mind. After decades of comparative neglect, Turing's 1936 paper "On Computable Numbers" is now regarded as the foundation stone of computability theory, and it is the fons et origo of the concept of computability employed in modern theoretical computer science. Moreover, Turing's 1950 essay, "Computing Machinery and Intelligence," sparked a rich literature on the mind-machine issue. Gödel's 1931 incompleteness results triggered, on the one hand, precise definitions of effective computability and its allied notions, and on the other, some much-criticized arguments for the conclusion that the mathematical power of the mind exceeds the limitations of Turing machines. Gödel himself is widely believed to have held that minds are more powerful than machines, whereas Turing is usually said to have taken the opposite position. In fact, neither of these characterizations is much more than a caricature. The actual picture is subtle and complex. To complicate matters still further, Gödel repeatedly praised Turing's analysis of computability, and yet in later life he accused Turing of fallaciously assuming, in the course of this analysis, that mental procedures cannot go beyond effective procedures. How can Turing's analysis be "unquestionably adequate" (Gödel 1964, 71) and yet involve a fallacy? We will present fresh interpretations of the positions of Turing and Gödel on computability and the mind. We argue that, contrary to first impressions, their views about computability are closer than might appear to be the case; and we will also argue that their views about the mind-machine issue are closer than Gödel-and others-have believed. 1 In section1.1, we show that Gödel's attribution of philosophical error to Turing is baseless; and we present a revisionary account of Turing's position regarding (what Gödel called) Hilbert's "rationalistic attitude." In section 1.2, we distinguish between two approaches to the analysis of computability, the cognitive and the noncognitive. We argue that Gödel pursued the noncognitive approach. As we will explain, we believe that Gödel mistook some cognitivist-style rhetoric in Turing's 1936 paper for an endorsement of the claim that the mind is

off at the next time step", and illustrates an interesting phenomenon: complex patterns on a large scale may emerge from simple computational rules on a small scale. If one were to look only at individual cells during the GL's computation, all one would see is cells switching on and off according to the four rules. Zoom out, however, and something else appears. Large structures, composed of many cells, grow and disintegrate over time. Some of these structures have recognizable characters: they maintain cohesion, move, reproduce, interact with each other. They are governed by their own rules. To discover these higher-order rules, one often needs to experiment, isolating the large structures and observing how they behave under various conditions. The behavior can be dizzyingly complex. Some patterns, consisting of hundreds of thousands of cells, behave like miniature universal Turing machines. Larger cellular patterns can build these universal Turing machines. Yet larger patterns feed instructions to the universal Turing machines to run GL. These in-game simulations of GL may themselves contain virtual creatures that program their own simulations, which program their own simulations, and so on. The nested levels of complexity that can emerge on a large grid are mind-boggling. Nevertheless, everything in GL is, in a pleasing sense, simple. The behavior of every pattern, large and small, evolves exclusively according to the four fundamental transition rules. Nothing happens in GL that is not determined by these rules. Zuse's thesis is that our universe is a computer governed by a small number of simple transition rules. Zuse suggested that, with the right transition rules, a cellular automaton would propagate patterns, which he called Digital-Teilchen (digital particles), that share properties with real particles. More recently, Gerard 't Hooft, Leonard Susskind, Juan Maldacena, and others have suggested that our universe could be a hologram arising from the transformation of digital information on a twodimensional surface (Bekenstein 2007). 't Hooft says: "I think Conway's Game of Life is the perfect example of a toy universe. I like to think that the universe we are in is something like this" ('t Hooft 2002). GL's four transition rules correspond to the fundamental "physics" of the GL universe. These are not the rules of our universe, but perhaps other transition rules are-or perhaps the universe's rules are those of some other type of computer: David Deutsch and Seth Lloyd suggest that the universe is a quantum-mechanical computer instead of a classical cellular automaton (Deutsch 2003, Lloyd 2006). If Zuse's thesis